We use mathematical models to study the mechanisms of oscillatory electrical activity arising from ion channels in cell membranes and modulated by intracellular chemical processes. We are interested in both the behavior of single cells and the ways in which cells communicate and modify each other's behavior. Our main application has been to the biophysical basis of insulin secretion in pancreatic beta-cells. We have examined bursting oscillations in membrane potential and the role of electrical coupling between cells in the islet of Langerhans. Long term goals are to understand how the membrane dynamics interact with intracellular events to regulate secretion. We also compare, contrast, and generalize to other secretory cells and neurons, including GnRH-secreting hypothalamic neurons, pituitary somatotrophs, and fast neurotransmitter secretion at nerve terminals. Our primary tool is the numerical solution of ordinary and partial differential equations. We use analytical, geometrical, graphical, and numerical techniques from the mathematical theory of dynamical systems to help construct and interpret the models. Perturbation techniques are used to get approximate analytical results in special cases. We study both detailed biophysical models and simplified models which are more amenable to analysis. Such an approach aids in the isolation of the essential or minimal mechanisms underlying phenomena, the search for general principles, and the application of concepts and analogies from other fields. Another role for our group is to mediate between the mathematical and biological disciplines. This includes disseminating the insights of mathematical work to biologists in accessible language and alerting mathematicians and other theoreticians to new and challenging problems arising from biological issues. Recent work on this project includes: 1. (Role of the Endoplasmic Reticulum in Shaping Calcium Oscillations) In the 2004 report we described in detail how a passive ER with linear uptake and release mechanisms is sufficient to account for most of the observed features of beta-cell cytosolic calcium. Active, nonlinear processes, such as calcium-induced calcium release are not required, and in fact would lead to behavior of cytosolic calcium in contradiction with observations. Systematic observation of ER calcium in beta-cells or islets during cytosolic calcium oscillations are still lacking, but the model predicts that ER calcium will oscillate in parallel. See Bertram and Sherman (2004). A review of the roles of cytosolic and ER calcium is in press. With the laboratory of Indu Ambudkar (NIDCR) we have developed a model of regulation of calcium entry by ER store content in a salivary gland cell line. The model proposes that entry is governed by a small sub-plasma-membrane compartment of the ER, rather than by the bulk ER. This is necessary to explain the triggering of entry by low doses of thapsigargin at early times when the bulk ER would not yet be substantially emptied. A paper is in review. 2. (Combined Electrical and Metabolic Oscillations in Pancreatic Islets) Although electrical oscillations in pancreatic islets are important for understanding many phenomena, their properties are at variance with observations of pulsatile insulin secretion in vivo. We have proposed that this can be explained by the modulation of the electrical oscillations by metabolic (glycolytic) oscillations. In particular, this combination can account for the observations of compound oscillations (bursts of bursts) that have been observed in membrane potential, cytosolic calcium, and metabolic variables such as intra-islet oxygen and glucose and mitochondrial membrane potential. We suggest that the glycolytic oscillations maintain optimal timing to coordinate insulin secretion and insulin action while the electrical oscillations control the quantity of insulin secreted in each puulse. See Bertram et al (2004). The model has provided an interpretation for experimental observations that islets from a given mouse are generally either fast or slow. We suggest that the slow mice have glycolytic oscillations, whereas the fast mice lack them. A paper is in press. A new framework for experimentation is opened up by these observations to test the model prediction and determine what factor(s) imprint the islets of a mouse. We have used the combined glycolytic-ionic model above to address the issues of how beta-cells synchronize within islets and how islets synchronize within the pancreas when metabolic oscillations are present. We found that the indirect calcium sensitivity of glycolysis allows pure electrical coupling to synchronize even the metabolic oscillations within islets, although including diffusion of the glycolytic intermediate fructose bis-phosphate makes synchrony more secure. For inter-islet synchrony, we found that the effect of insulin secretion into the common circulation is sufficient for synchrony by entraining the islets to a common glucose input. See Pedersen et al (2005). We have also found that diffusion of calcium or the glycolytic intermediate glucose-6-phosphate between beta-cells can kill the oscillations rather than enhancing synchrony. This may account for the observation that over-expression of gap junction proteins can convert slow (presumably glycolytically driven) calcium oscillations into fast (presumably purely ionic) calcium oscillations. Experimental tests of this prediction are planned and two papers are in preparation. 3. (Adipocyte cell size distributions) Using data supplied by the Cushman (NIDDK) and Reaven (Stanford) labs we have carried out a statistical analysis of cell size distributions in obese human subjects who were either insulin resistant or insulin sensitive. Detailed histograms of cell diameters for approximately 5000 cells per sample were obtained using a Coulter counter. The statistical analysis revealed that the insulin senstive patients had a greater proportion of large cells, the only subpopulation of cells capable of significant fat storage. This was surprising in view of evidence that large cells are more insulin resistant on a per cell basis than small cells, but is not in contradiction to those data. We suggest that insulin resistant subjects have a defect in adipocyte recruitment or differentiation that prevents them from matching fat storage capacity to demand. A paper is in preparation.